Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1

Average Error: 2.2 → 2.0

Time: 3.9s

Precision: binary64

Math

\[\frac{x}{y} \cdot \left(z - t\right) + t \]

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4152352409574924 \cdot 10^{-67}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z - t}{y}, t\right)\\ \mathbf{elif}\;y \leq -1.446115896866537 \cdot 10^{-282}:\\ \;\;\;\;t + \frac{x \cdot \left(z - t\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t + \left(z - t\right) \cdot \frac{x}{y}\\ \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ (* (/ x y) (- z t)) t))

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.4152352409574924e-67)
   (fma x (/ (- z t) y) t)
   (if (<= y -1.446115896866537e-282)
     (+ t (/ (* x (- z t)) y))
     (+ t (* (- z t) (/ x y))))))

C

double code(double x, double y, double z, double t) {
	return ((x / y) * (z - t)) + t;
}

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.4152352409574924e-67) {
		tmp = fma(x, ((z - t) / y), t);
	} else if (y <= -1.446115896866537e-282) {
		tmp = t + ((x * (z - t)) / y);
	} else {
		tmp = t + ((z - t) * (x / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x / y) * Float64(z - t)) + t)
end

↓

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.4152352409574924e-67)
		tmp = fma(x, Float64(Float64(z - t) / y), t);
	elseif (y <= -1.446115896866537e-282)
		tmp = Float64(t + Float64(Float64(x * Float64(z - t)) / y));
	else
		tmp = Float64(t + Float64(Float64(z - t) * Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]

↓

code[x_, y_, z_, t_] := If[LessEqual[y, -2.4152352409574924e-67], N[(x * N[(N[(z - t), $MachinePrecision] / y), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[y, -1.446115896866537e-282], N[(t + N[(N[(x * N[(z - t), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(t + N[(N[(z - t), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.2
Target	2.3
Herbie	2.0

\[\begin{array}{l} \mathbf{if}\;z < 2.759456554562692 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z < 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if y < -2.41523524095749241e-67

   1. Initial program 1.2

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]

   2. Simplified 1.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{z - t}{y}, t\right)} \]

   if -2.41523524095749241e-67 < y < -1.44611589686653695e-282

   1. Initial program 4.1

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]

   2. Taylor expanded in x around 0 1.4

      \[\leadsto \color{blue}{\frac{\left(z - t\right) \cdot x}{y}} + t \]

   if -1.44611589686653695e-282 < y

   1. Initial program 2.5

      \[\frac{x}{y} \cdot \left(z - t\right) + t \]
3. Recombined 3 regimes into one program.

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4152352409574924 \cdot 10^{-67}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z - t}{y}, t\right)\\ \mathbf{elif}\;y \leq -1.446115896866537 \cdot 10^{-282}:\\ \;\;\;\;t + \frac{x \cdot \left(z - t\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t + \left(z - t\right) \cdot \frac{x}{y}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"
  :precision binary64

  :herbie-target
  (if (< z 2.759456554562692e-282) (+ (* (/ x y) (- z t)) t) (if (< z 2.326994450874436e-110) (+ (* x (/ (- z t) y)) t) (+ (* (/ x y) (- z t)) t)))

  (+ (* (/ x y) (- z t)) t))